Hanada, Takao

Ishimura, Naoyuki

Nakamura, Masaaki

Kyoto University Research Information Repository

TitleSingular perturbation problem for a model equation of phaseseparation (Dynamics of spatio - temporal patterns for thesystem of reaction - diffusion equations)Author(s)Ishimura, Naoyuki; Nakamura, Masaaki; Hanada, TakaoCitation数理解析研究所講究録 (2005), 1416: 85-93Issue Date2005-02URL http://hdl.handle.net/2433/26271RightType Departmental Bulletin PaperTextversionpublisherKyoto University85Singular perturbation problem foramodel equation of phase separation( ) (ISHIMURA, Naoyuki)( ) (NAKAMURA, MasaAki)( ) (HANADA, Takao)The behavior of steady state solutions of amodel equation forphase separation is analyzed when the parameter, which dependson the ratio of the surface energy, tends to zero. Analytical con-vergence results are exhibited.1IntroductionPhase separations observed in avariety of material sciences are fascinat-ing topics for research both from the mathematical and computationalpoint of view [14]. The motion law introduced by Eguchi, Oki, and Mat-sumura (EOM), is derived from the first principles of thermodynamics ofirreversible process; the system extends and connects the Cahn-Hilliardequation [1] and the Ginzburg-Landau equation. In one space dimension,under asuitable scaling of parameters, the EOM equation is expressed asfollows [9] [10].ut=-\epsilon 2ux xx+((a+v2)u\sim x$ut=-e2uxxx +((a+v )xx$ in $0<x<l$ , $t>0$$v_{t}=v_{xx}+(b-u^{2}-v^{2})v$ in $0<x<l$ , $t>0$(1)$u_{x}=u_{xxx}=v_{x}=0$ at $x=0$ and 1, $t>0$$u|_{t=0}=u_{0}$ , $v|_{t=0}=v_{0}$ on $0\leq x\leq l$ ,where $u$ . $v$ denote the local concentration and the local degree of order,respectively. The constants $a$ , $b$ are assumed to be positive. The param-eter $\epsilon$ depends on the ratio of the surface energy between the original$u$ and $v$ . It is easily noticed that the equation for $u$ is Cahn-Hilliardtype while the one for $v$ is Ginzburg-Landau type. Adirect computationshows that the total concentration of $u$ is conserved under the evolution1416 2005 85-9386namely, there holds$\frac{1}{l}\int_{0}^{l}u(x, t)dx=m$ , (2)where $m$ is a positive constant. Given initial data $u_{0}$ , $v_{0}$ should satisfyrequired compatibility conditions:$(u_{0})_{x}=(u_{0})_{xxx}=(v_{0})_{x}=0$ at $x=0$ , $l$ , and $\frac{1}{l}\int_{0}^{l}u_{0}(x)dx=m$ .For further information and other related issues, especially for the studyof Cahn-Hilliard equations, we refer to $[2][3][5][6][7][11][12][13][15]$ andthe references cited therein. We here just mention that the system (1)takes its rise in a gradient flow of the total free energy$F_{\epsilon}[u, v]:= \int_{0}^{l}(\frac{\epsilon^{2}}{2}u_{x}^{2}+\frac{1}{2}v_{x}^{2}+\frac{a}{2}u^{2}+\frac{1}{4}v^{4}-\frac{b}{2}v^{2}+\frac{1}{2}u^{2}v^{2})dx$ . (3)Indeed one easily show that$\frac{d}{dt}F_{\epsilon}[u, v](t)=-\int_{0}^{l}\{-\epsilon^{2}u_{xxx}+((a+v^{2})u)_{x}\}^{2}dx-\int_{0}^{l}v_{t}^{2}dx\leq 0$ ,for any solution $(u, v)$ to (1); $F_{\epsilon}$ plays the role as Lyapunov functionalof the system. Here and hereafter., the solution are understood to beclassical.In this article, we deal with the singular perturbation problem for (1).In particular, we are concerned with the behavior of steady state solutionsas the parameter $\epsilon$ tends to 0. We recall that steady state solutions aregiven by $u_{t}=v_{t}\equiv 0$ in (1); that is,$\{$$-\epsilon^{2}u_{xxxx}+((a+v^{2})u)_{xx}=0$ in $0<x<l$$v_{xx}+(b-u^{2}-v^{2})v=0$ in $0<x<l$$u_{x}=u_{xxx}=v_{x}=0$ at $x=0$ and $l$$(1/l) \int_{0}^{l}udx=m$ .(4)In [9], it is shown that every solution of (1) converges to a steady statesolution as $tarrow\infty$ . Furthermore, the solution $(u, v)$ to the system (4) isconstructed as a critical point of the functional (3) in the function space$A:= \{(u, v)\in(H^{1}(0, l))^{2}|\frac{1}{l}\int_{0}^{l}udx=m\}$ .87We recall that $(u, v)=(m, 0)$ and $(m, \pm\sqrt{b-m^{2}})$ if $b>m^{2}$ are refered toas trivial solutions. One of main achievements in [9] is that the structureof steady state solutions is rather rich enough; to be specific, the nextproperties is established in $[8][9]$ .Theorem 1 ([9]) There is at least one monotone non-trivial steady statesolution of (4) if we assign suitably large $b$ and $m^{2}$ . Moreover, for anyinteger $k\geq 2$ and for appropriately chosen large $b$ and $m^{2}$ depending on$k$ , (4) has at least one non-monotone non-trivial steady state solution,each of whose derivatives changes sign exactly $(k-1)$ -times.It is a challenging problem to study the behavior as $\epsilon$ $arrow 0$ , $tarrow\infty$ forsteady state solutions $(u, v)$ . This is the focus of this paper.If we put $\epsilon \mathrm{i}$ $=0$ in (4) then we find that$(a+v^{2})u=M$ (5)for some constant $M$ . Invoking (2) we compute $\tau n$ $=l^{-1} \int_{0}^{l}M/(a+v^{2})dx$and hence$M= \frac{lm}{\int_{0}^{l}(a+v^{2})^{-1}dx}>0$ . (6)Therefore the reduced equation of $\epsilon$ $=0$ for $v$ is described as$\{$$v_{xx}+(b-( \frac{M}{a+v^{2}})^{2}-v^{2})v=0$ in $0<x<l$$v_{x}=0$ at $x=0$ and $l$ ,(7)where $M$ is defined by (6). The equations $(5)(6)(7)$ is analysed in \S 2.Our intension is now to ask whether the solutions of (4) “converge” $\mathrm{t}\mathrm{t}$)the ones of (7) as $\epsilon$ $arrow 0$ or not? If this really holds true, in what spacesor norms do they converge? Our answer to this question reads as follows.Theorem 2 Let $(u^{\epsilon}, v^{\epsilon})$ denote solutions to (4) with positive parameters$\epsilon \mathrm{i}$ . The constants $a$ , $b$ , $m$ are kept fixed. Then there exists a monotonedecreasing sequence $\{\epsilon_{n}\}_{n=1}^{\infty}$ tending to zero such that$u^{\epsilon_{n}}arrow u$ strongly in $L^{2}(0, l)$(8)$v^{\epsilon_{n}}arrow v$ in $C[0, l]\cap C^{1}(0, l)$as $narrow\infty$ , where $(u, v)$ solves the reduced equations $(\mathit{5})(\mathit{6})(7)$ .882 Reduced equationsIn this section, we investigate the reduced equation (7) with (6), whichis provided by assigning $\Xi$ $=0$ in (4). As is carried out in our previ-ous works $[8][9]$ , a variational method is exploited. We recall the func-tional (3) with $\epsilon=0$ over the function space $A’:=\{(u, v)\in L^{2}(0, l)\cross$$H^{1}(0, l)|l^{-1}J_{0}^{\backslash l}udx=m\}\subset A$ ; that is,$F_{0}[u, v]:=. \int_{0}^{l}(\frac{1}{2}v_{x}^{2}+\frac{a}{2}u^{2}+\frac{1}{4}v^{4}-\frac{b}{2}v^{2}+\frac{1}{2}u^{2}v^{2})dx$ .For any $\psi$ $\in L^{2}(0, l)$ with $\int_{0}^{l}\psi$ $dx=0$ and $\varphi\in H^{1}(0, l)$ , we derive$\frac{d}{d\sigma}F_{0}[(u+\sigma\psi)v]|_{\sigma=0}=\int_{0}^{l}(au+v^{2})\acute{\psi}dx$,$\frac{d}{d\eta}F_{0}[u, v+\eta\varphi]|_{\eta=0}=f_{0}^{l}\{v_{x}\varphi_{x}+(v^{2}‘+u^{2}-b)v\varphi\}dx$,which immediately implies that every critical point of $F_{0}$ over $A’$ givesrise to the solutions (5) (7).Taking account that the energy level of trivial solutions are$F_{0}[m, 0]= \frac{lm^{2}a}{2}$ , $F_{0}[m, \pm\sqrt{b-m^{2}}]=\frac{lm^{2}a}{2}-\frac{l}{4}(b-m^{2})^{2}$ if $b>m^{2}$ .we $\mathrm{n}$ eed to seek a test function $(\psi, \varphi)\in A’$ such that$F_{0}[ \psi, \varphi]<\frac{lm^{2}a}{2}-\frac{l}{4}(b-m^{2})^{2}$ . (9)$r\Gamma \mathrm{t}\iota \mathrm{e}\mathrm{n}$ the standard minimization procedure yields a non-trivial solutionfor $(5)(7)$ . To this aim, as performed in our previous paper [9], we intr0-duce$\psi(x)=m-\delta\cos\frac{\pi x}{l}$ , $\varphi(x)=\pm\sqrt{b-(m-\delta\cos\frac{\pi x}{l})^{2}}$ ,where $\delta>0$ with $b>$ $(rn +\delta)^{2}$ is determined later. We compute$F_{0}[ \psi, \varphi]-\{\frac{lm^{2}a}{2}-\frac{l}{4}(b-m^{2})^{2}\}$$= \frac{la\delta^{2}}{4}+\frac{\delta^{2}\pi^{2}}{2l}\int_{0}^{l}\frac{(m-\delta\cos(\tau_{\mathfrak{l}}x/l))^{2}\sin(\pi x/l)}{b-(m-\delta\cos(\pi x/l))^{2}}dx$$- \frac{lm^{2}\delta^{2}}{2}-\frac{3l}{32}\delta^{4}+\frac{l\delta^{2}}{4}(b-m^{2})$ .89If we further assume that $m^{2}=b/2$ and $\delta=\sqrt{b}/4$ , then we infer that$F_{0}[ \psi, \varphi]-\{\frac{lm^{2}a}{2}-\frac{l}{4}(b-m^{2})^{2}\}$$=- \frac{67}{2^{13}}lb^{2}+(\frac{la}{64}+\frac{\pi^{2}}{32l}\int_{0}^{l}\frac{(4-\sqrt{2}\cos(\pi x/l))^{2}\sin(\pi x/l)}{32-(4-\sqrt{2}\cos(\pi x/l))^{2}}dx)b$,from where we obtain (9) choosing larger $b$ if necessary. The monotonicityof $v$ as well as the existence of multi-bump solutions can be accomplishedsimilarly as in [9]. To summarize we have 1)roved the next proposition.Proposition 3 For suitably large $b$ and $m^{2}$ , there exists at least onemonotone non-trivial solution of $(\mathit{5})(7)$ . Moreover, for any integer $k\geq 2$and for appropriately chosen large $b$ and $m^{2}$ depending on $k$ , $(\mathit{5})(7)$ hasat least one non-monotone non-trivial solution, whose derivative changessign exactly $(k-1)$ times, respectively.3 Proof of the convergence theoremThis section is devoted to $\mathrm{t}1_{1}\mathrm{e}$ proof of Theorem 2. We begin with a prioriestimates. An integration yields that the system (4) is eqllival(\,nt to$\{$$-\cup-^{2}u_{xx}+(a+\uparrow j^{2})u=M_{\epsilon}$ in $0<x<l$$v_{xx}+(b-u^{2}-v^{2})v=0$ in $0<x<l$$u_{x}=v_{x}=0$ at $x=0$ and $l$$(1/l) \int_{0}^{l}udx=m$ ,(10)for certain constant $M_{\epsilon}$ . The next proposition $\mathrm{i}\mathrm{s}^{\backslash }$ in order, where $\mathrm{t}\mathrm{I}_{1}\mathrm{e}$solution of (10) is simply written as $(u, v)$ instead of $(u^{\epsilon}, v^{\epsilon})$ , withoutpossible confusion.Proposition 4 For every solution $(u, v)$ of (10), it follows that. inde-pendent of $\epsilon$ ,$\frac{a}{a+m}m\leq u(x)\leq\frac{a+b}{a}m$ on $0\leq x\leq l$ ,$\max_{x\in[0,l]}|v(x)|\leq\sqrt{b}$ , $am\leq NI_{\epsilon}\leq(a+b)m$ .90Furthermore, there exists a constant $C$ independent of $\epsilon$ such that$\int_{0}^{l}\epsilon^{2}u_{x}^{2}dx+\int_{0}^{l}v_{x}^{2}dx\leq C$.Proof. First we deal with a bound on $v$ . If there happens to be $v(x)>\sqrt{b}$over some interval, then at the maximum point $x_{M}$ of $v$ there holds$0\geq v_{xx}(x_{M})=v(x_{M})(v(x_{M})^{2}+u(x_{M})^{2}-b)>0$,a contradiction. The chance of being $v(x)<-\sqrt{b}$ is similarly discardedand we have proved $\max|v|\leq\sqrt{b}$ .Next, at the minimum point $x_{m}$ of $u$ , we see that$0\geq-\epsilon^{2}u_{xx}(x_{m})=M_{\Xi}-(a+v(x_{m})^{2})u(x_{m}\mathrm{I}j$$x \in[0,l]\mathrm{r}\mathrm{n}\mathrm{i}\mathrm{n}u(x)=u(x_{m})\geq\frac{M_{\in}}{a+v(x_{m})^{2}}\geq\frac{M_{\epsilon}}{a+b}$ ,from where we obtain$m= \frac{1}{l}\int_{0}^{l}udx\geq\frac{M_{\in}}{a+b}$ .A similar consideration at the maximum point of $u$ leads us to$\max_{x\in[0,l]}u(x)\leq\frac{M_{F}}{a+v^{2}}-\leq\frac{M_{\epsilon}}{a}\leq\frac{a+b}{a}m$,$m= \frac{1}{l}J_{0}^{l}udx\leq\frac{\mathrm{J}/I_{\epsilon}}{a}$ .Finally we ascertain integral estimates. Multiply the equations in (10)by $u$ and $v$ , respectively, and integrate, we deduce that$\int_{0}^{l}\epsilon^{2}u_{x}^{2}dx+\int_{0}^{l}(a+v^{2})u^{2}dx=\int_{0}^{l}M_{\epsilon}udx=lmM_{\mathit{6}}$ ,$\int_{0}^{l}v_{x}^{2}dx=\int_{0}^{l}(b-u^{2}-v^{2})v^{2}dx$ .Combined with pointwise estimates above, we have finished the derivationof integral bounds91Now we complete the proof Theorem 2. Wc appeal to the variation ofconstants formula.$\epsilon^{2}u^{\epsilon}(x)=\frac{1}{2}\int_{0}^{x}\{(a+v^{\in}(y)^{2})u^{\epsilon}(y)-M_{\epsilon}\}(x-y)dy$$+ \frac{1}{2}\int_{x}^{l}\{(a+v^{\epsilon}(y)^{2})u^{\epsilon}(y)-M_{\Xi}\}(y-x)dy$(11)$v^{\Xi}(x)= \frac{1}{2}\int_{0}^{x}(v^{\epsilon}(y)^{2}+u^{\epsilon}(y)^{2}-b)v^{\epsilon}(y)(x-y)dy$$+ \frac{1}{2}\int_{x}^{l}(v^{\epsilon}(y)^{2}+u^{\epsilon}(y)^{2}-b)v^{\epsilon}(y)(y-x)dy$ .Thanks to Proposition 4, we infer that a monotone decreasing sequence$\epsilon_{1}$ $>\epsilon_{2}>|$ $\cdot\cdot>\epsilon_{n}>\cdotsarrow 0$ can be selected so that $M_{\sigma}arrow M$ as well as$u^{\epsilon_{n}}arrow u$ , $v^{\epsilon_{n}}arrow v$ strongly in $L^{2}(0, l)$ ,where $am\leq M\leq(a+b)m$ means for a constant and $u$ , $v\in L^{2}(0, l)$ arcbounded functions. Sending $\xi jn$ $arrow 0$ in (11), wc conclude that$0= \frac{1}{2}\int_{0}^{x}\{(a+v(y)^{2})u(y)-M\}(x-y)dy$$+ \frac{1}{2}\int_{x}^{l}\{(a+v(y)^{2})u(y)-M\}(y-x)dy$$v(x)= \frac{1}{2}\int_{0}^{x}(v(y)^{2}+u(y)^{2}-b)v(y)(x-y)dy$$+ \frac{1}{2}\int_{x}^{l}(v(y)^{2}+u(y)^{2}-b)v(y)(y-x)dy$ .which implies that $(u, v)$ solves (5) (7). This finalizes the proof of Theorem2.4 $\mathrm{D}_{\acute{1}}$scussionsWe investigate a singular perturbation problem for steady state solutionsto a model equation of phase separation. Analytical convergence resultsare established as the parameter, which depends on the surface energybetween the components, tends to zero. It is clarified that the rich struc-ture of steady state solutions is preserved under this singular perturbationlimit92Acknowledgements. This work is partially supported by Grants-in-Aidsfor Scientific Research $(_{[perp]}^{\backslash }\mathrm{J}\mathrm{o}\mathrm{s}. 13555021, 15634006, 16540184)$, from theJapan Society for Prom otion of Sciences.$\#_{arrow\vee}\yen \mathrm{X}\mathbb{R}^{\mathrm{t}}$[1] $\mathrm{J}.\mathrm{W}$ . Cahn and $\mathrm{J}.\mathrm{E}$ . Hilliard; Free energy of a nonuniform system,I., Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.[2] J. Carr, $\mathrm{M}.\mathrm{E}$ . Gurtin, and M. Slemrod; Structured phase transitionson a finite interval, Arch. Rational Mech. Anal, 86 (1984), 317-351.[3] X. Chen; Global asymptotic limit of solutions of the Cahn-Hilliardequation, J. Differential Geometry, 44 (1996), 262-311.[4] $\mathrm{T}$ Eguchi, K.Oki, and S. Matsumura; Kinetics of ordering withphase separation, in “Phase Transformations in Solids, Mat. ${\rm Res}$ .Soc. Symp. Proc. 21,” T. 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Temam; Infinite-Dimensional Dynamical Systems in Mechanicsand Physics, Applied Mathematical Sciences 68, Springer-Verlag,New York, 1988.[15] Zheng S. ; Asymptotic behavior of solution to the $\mathrm{C}\mathrm{a}1_{\mathrm{l}}\mathrm{n}$-Hilliardequation, Applicable Anal., 23 (1986), 165-184

Singular perturbation problem for a model equation of phase separation (Dynamics of spatio - temporal patterns for the system of reaction - diffusion equations)

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Singular perturbation problem for a model equation of phase separation (Dynamics of spatio - temporal patterns for the system of reaction - diffusion equations)

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